Research Topic in Graph Theory or Non-Well-Founded Set Theory

I'm doing to come up with a subject in either of them to do either an "independent study" or "project" on, the former is a course which simply requires you to learn the subject and the latter is "independent study" + a x-page paper. Unfortunately I don't know either subject too well so I can't come up with anything specific enough. Can anyone make a suggestion?

non well-founded set theory and use it to build up the foundation for non-standard analysis because I found graph theory to be sort of boring. But, some think non-standard analysis is virtually useless as history has never provided a situation when a theorem in standard analysis could only be proven by non-standard analysis techniques. Nevertheless, it would be cool to really understand it if you have an interest in analysis and if you have a interest in mathematical logic then this is also obviously a good choice (but in that regard so is graph theory as it is an example of descriptive set theory). As far as I know one of the most basic objects in model theory is an Ultrafilter and I am sure you would get a really good understanding of these in non well-founded set theory. Depends on your other interest really.

That's the problem with non-well-founded set theory: nobody in my department seems to care, or want to do it. The problem with graph theory though is that since I know very little of it, the questions I have are either well-known hard problems or easy to solve problems, nothing in the middle that I could work on.

well, if you ask for material on non well founded set theory, then there's jonh barwise's vicoious circle for first glance (you can view it for free from stanford), and you can search tom forster from cambridge university, I think he's a major reasercher is in this field.

I can't find the book by Barwise that you said. Are you sure you have the title right?

I've decided to do it on non-well-founded set theory (my other alternative is Fourier Analysis, but I decided against it). Unfortunately there is only ONE book which deals with the subject, namely "Non-Well-Founded sets" by Azcel. Does anyone know of other books (preferably textbooks) which deal with the subject?